Of mice and chickens: Revisiting the RC time constant problem

Abstract

Avian hair cells depend on electrical resonance for frequency selectivity. The upper bound of the frequency range is limited by the RC time constant of hair cells because the sharpness of tuning requires that the resonance frequency must be lower than the RC roll-off frequency. In contrast, tuned mechanical vibration of the inner ear is the basis of frequency selectivity of the mammalian ear. This mechanical vibration is supported by outer hair cells (OHC) with their electromotility (or piezoelectricity), which is driven by the receptor potential. Thus, it is also subjected to the RC time constant problem. Association of OHCs with a system with mechanical resonance leads to piezoelectric resonance. This resonance can nullify the membrane capacitance and solves the RC time constant problem for OHCs. Therefore, avian and mammalian ears solve the same problem in the opposite way. This article is part of the Special Issue Outer hair cell Edited by Joseph Santos-Sacchi and Kumar Navaratnam.

Keywords: Auditory frequencies; Electrical resonance; Membrane capacitance; Piezoelectric resonance.

Figure A.1:

The effect changing anti-drag to drag. The sign of ${\eta }_1$ in Eq. 9a is changed from minus to plus. A: Normalized displacement $\left|X/X_0\right|$ of mass $M$ at the resonance frequency ${\omega }_0$, is plotted against the mass ratio $m/M$ and the reduced drag coefficient $\overline{\eta }=\eta /\left(M{\omega }_0\right)$. The normalization factor is $X_0=f_0/K$. B: The frequency dependence at $\overline{\eta }=0.11$. Traces are for $m/M=0.1$, 0.15, and 0.2 from the left. The abscissa is the normalized frequency $\omega /{\omega }_0$.

Figure 1:

An equivalent circuit for electric resonance. $r_a$ is hair bundle resistance and $R_m$ is cell body resistance. An interplay between $\text{C}{\text{a}}^{2+}$-channel and $\text{C}{\text{a}}^{2+}$-activated ${\text{K}}^{+}$ channel [20, 23] leads to electrical resonance, which can be represented by a combination of inductance $L$ and capacitance $C_m$ [19].

Figure 2:

The effect of resonance frequency and RC roll-off frequency on the amplitude $\nu$ of the receptor potential (Eq. 8). The amplitude $v$ is normalized at $\omega =0$. A: The effect of changing the resonance the ratio ${\omega }_{rc}$. The frequency axis is normalized by the RC roll-off frequency ${\omega }_{rc}$. The values for ${\omega }_r/{\omega }_{rc}$ are 0.5, 0.7, 0.9, 1.1, and 1.3 (red arrows). The broken line shows the response without resonance. The value of $R$ is set to unity. B: The effect of the membrane resistance. Notice the peak shift to lower frequencies as well as the broadening as the resistance increases. The frequency axis is normalized by the resonance frequency. Relative values of the membrane resistance are indicated. Here $“R”$ represents the condition ${\omega }_{rc}/{\omega }_r=1.5$. C: The phase response that corresponds to B. The phase is approximately constant near $\omega /{\omega }_r=1$.

Figure 3:

The cross-section of the avian inner ear. TM: the tectorial membrane, BM: the basilar membrane, BP: the basilar papilla. After refs. [22, 26].

Figure 4:

A possible mode of motion of the avian inner ear. Two squares represent two parts of the tectorial membrane. The smaller one on top of SHCs and the larger one on top of THCs. The tapering the tectorial membrane is represented by the mass ratio $M/m$.

Figure 5:

The effect of the mass ratio. A: Normalized displacement $\left|X/X_0\right|$ of mass $M$ at the resonance frequency ${\omega }_0$, is plotted against the mass ratio $m/M$ and the reduced drag coefficient $\overline{\eta }=\eta /\left(M{\omega }_0\right)$. The normalization factor is $X_0=f_0/K$. B: The frequency dependence at $\overline{\eta }=0.11$. Traces are for $m/M=0.1$, 0.15, and 0.2 from the left. The abscissa is the normalized frequency $\omega /{\omega }_0$. The half-peak frequency $\text{Δ}\overline{\omega }$ for the peak of $m/M=0.1$ is 0.05, indicating the quality factor of 20.

Figure 6:

OHCs incorporated into a mechanically resonating system. $K_e$ is the stiffness of the elastic load, $\eta$ the drag coefficient, and $m$ the inertial mass. The intrinsic stiffness of the OHC is $k_s$.

Figure 7:

Frequency dependence of the membrane capacitance and the amplitude of the receptor potential of OHC. The frequency axis is the reduced frequency $\overline{\omega }\left(=\omega /{\omega }_r\right)$, normalized by the frequency ${\omega }_r$ of mechanical resonance. A: Motile charge produces negative capacitance near resonance frequency, eliminating the structural membrane capacitance $C_0$. At the frequencies of maximal power generation by an OHC, which are marked by filled red circles, the total membrane capacitance is slightly positive, as indicated by Eq. 21. B: The receptor potential resulting from the frequency dependence of the capacitance. The traces are color coded, matching with those in A. The normalization factor $v_0$ is given by $v_0=i_{0}r/{\left(\omega C_0\right)}^2$. The traces are: $\zeta =1$ (red), 0.52 (blue), and 0.24 (black).